We describe an algorithm for minimizing convex, not necessarily smooth, functions of several variables, based on a descent direction finding procedure that inherits some characteristics both of standard bundle method and of Wolfe’s conjugate subgradient method. This is obtained by allowing appropriate upward shifting of the affine approximations of the objective function which contribute to the classic definition of the cutting plane function. The algorithm embeds a proximity control strategy. Finite termination is proved at a point satisfying an approximate optimality condition and some numerical results are provided.